Page 27 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I reAl numBers

Solution: Let the minimum positive fraction be y/x , where x, y are coprime positive
y 54 y 175 y 55
integers, then ÷ = × and × are both integers. Thus 175/54 and 55/36
x 175 x 54 x 36
are irreducible fractions, then x is a common divisor of 175 and 55, and y is the smallest

common multiple of 54 and 36. To minimize y/x , we should maximize x and minimize

y , then x should be the largest common divisor of 175 = 5 × 7 and 55 = 5 × 11, which
2
is 5, and y should be the smallest common multiple of 54 = 2 × 3 and 36 = 2 × 3 ,
2
3
2
which is 2 × 3 = 108. Therefore, the minimum positive fraction y/x = 108/5.
3
2
1.72 a, b, c, d are positive integers, and a = b ,c = d ,c − a = 11, evaluate d − b .
5
4
3
2
2 0
Solution: Let a = b = t 0 where t is a positive integer, then a = t ,b = t . Let c = d = p
5
3
2
6
4
4
5
where p is a positive integer, then c = p ,d = p . In addition, c − a = 11 , then
2
3
√
2
2
2
2
2
4
p − t = 11 ⇒ (p − t )(p + t ) ⇒ p − t =1,p + t = 11 ⇒ p =6,t = √ 5,b =
2
2
2
2
2
4
√ p − √ t = 11 ⇒ (p − t )(p + t ) ⇒ p − t =1,p + t = 11 ⇒ p =6,t = 5,b =
√
3
5 √
√
√
( 5) = 25 5,d =6 = 216,d − 6 = 216 − 25 5
( 5) = 25 5,d =6 = 216,d − 6 = 216 − 25 5 .
5
3
1.73 Given x + y − z = x − y + z = y + z − x and xyz =0, evaluate (x + y)(y + z)(z + x) .
z y x xyz
Solution: Let x + y − z = x − y + z = y + z − x = k, then x + y − z = kz (i), x − y + z = ky
z y x
(ii), y + z − x = kx (iii). (i)+(ii)+(iii): x + y + z = k(x + y + z) ⇒ (k − 1)(x + y + z)=0. There
are two possibilities k =1 or x + y + z =0. When k =1, then x + y =2z, x + z =2y, y + z =2x,
then (x + y)(y + z)(z + x) = 2z · 2x · 2y =8 . When x + y + z =0 , then
xyz xyz
x + y = −z, y + z = −x, z + x = −y , then (x + y)(y + z)(z + x) = (−z) · (−x) · (−y) = −1 . As a
xyz xyz
conclusion, (x + y)(y + z)(z + x) is 8 or -1.
xyz
1.74 Let

1 1 1 1 1 1 1 1
S = 1+ + + 1+ + + 1+ + + ··· + 1+ + , find the integer
1 2 2 2 2 2 3 2 3 2 4 2 2010 2 2011 2
part of S .

Download free eBooks at bookboon.com
27